IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 15, NO. 2, MAY 2000 735

Robust Tuning of Power System Stabilizers inMultimachine Power Systems

Y. L. Abdel-Magid, Senior Member, IEEE, M. A. Abido, Member, IEEE, and A. H. Mantawy, Member, IEEE

Abstract—This paper demonstrates the robust tuning of powersystems stabilizers for power systems, operating at differentloading conditions. A classical lead-lag power system stabilizeris used to demonstrate the technique. The problem of selectingthe stabilizer parameters is converted to a simple optimizationproblem with an eigenvalue-based objective function, which issolved by a tabu search algorithm. The objective function allowsthe selection of the stabilizer parameters to optimally place theclosed-loop eigenvalues in the left-hand side of a vertical line inthe complex -plane. The effectiveness of the stabilizers tunedusing the suggested technique, in enhancing the stability of powersystems, is confirmed through eigenvalue analysis and simulationresults.

Index Terms—Low frequency oscillation, Tabu search, Powersystem stability, Robust control.

I. INTRODUCTION

T HE application of tabu search (TS) algorithms has recentlyattracted the attention of researchers in the field of ar-

tificial intelligence. From the literature, it is clearly seen thattabu search algorithms can provide powerful tools for optimiza-tion [1]–[4]. Tabu search is an iterative improvement procedurewhose main advantage is its ability to escape local optima. Tabusearch is independent of the complexity of the objective func-tion considered. It suffices to specify the objective function andto place finite bounds on the parameters.

The importance of increasing the stability boundaries of syn-chronous machines equipped with fast-acting static exciters isvery well known. Several techniques have been used for thedesign of supplementary excitation controllers for this purpose[5]–[19].

It is well known that machine parameters change withloading, making the dynamic behavior of the machine quitedifferent at different operating conditions [6]–[8]. Conse-quently, a set of power system stabilizer [PSS] parameterswhich stabilizes the system under a certain operating conditionmay no longer yield satisfactory results when there is a drasticchange in the operating point.

In this paper, the power system, operating at various loadingconditions is treated as a finite set of plants, and a robust PSSwith parameters that can simultaneously improve the dampingof this set of plants is determined off-line using a tabu searchand an objective function based on the system eigenvalues. The

Manuscript received October 19, 1998; revised June 1, 1999. This work wassupported by King Fahd University of Petroleum and Minerals.

The authors are with the Electrical Engineering Department, King Fahd Uni-versity of Petroleum and Minerals, Dhahran 31261, Saudi Arabia (e-mail: {am-agid; mabido; amantawy}@kfupm.edu.sa).

Publisher Item Identifier S 0885-8950(00)03812-8.

PSS designed in this manner will perform well under variousloading conditions and stability of the system is guaranteed. Bycontrast, a PSS designed for a certain operating point will onlyperform optimally at the design loading condition.

The suggested eigenvalue-based objective function will opti-mally place the closed-loop eigenvalues of the power system inthe left-hand side of a vertical line in the complex-plane. Asingle machine infinite bus system, and a multimachine powersystem are used to demonstrate the suggested technique

The TS can easily incorporate most types of constraints andstructures on the controller and often leads to the global op-timum [1]. Moreover, the problem of simultaneous eigenvalueplacement, whose analytic solution is yet unknown, is equallyand easily solved using TS as shown in this work.

Simulation results and eigenvalue analysis are usedthroughout the paper to assess the effectiveness of the tech-nique suggested in this work.

II. TABU SEARCH

Tabu Search was originally proposed as an optimization toolby Glover in 1977 [1]–[4]. It is a conceptually simple and anelegant iterative technique for finding good solutions to opti-mization problems. In general terms, TS is characterized by itsability to escape local optima by using a short-term memory ofrecent solutions called the tabu list. Moreover, tabu search per-mits backtracking to previous solutions by using the aspirationcriteria.

Tabu search is an iterative improvement procedure in thatit goes from some initial feasible trial solution to another bymaking moves. It makes several candidate moves and selects themove producing the best solution among all candidate movesfor the current iteration. The set of admissible moves forms acandidate list. The best candidate solution becomes the currentsolution. It is possible that the tabu search may visit the samesolution again in latter iterations. Thus, there is a possibility ofcycling. Tabu restriction is a means to avoid such cycling bymaking the current solution tabu i.e., forbidden. Tabu restric-tions are enforced by a tabu list. The tabu list is a list containingforbidden moves (solutions). It is forbidden to accept the samesolution as long as it is in the tabu list. Tabu restrictions allowthe search to go beyond the points of local optimality while stillmaking the best possible move in each iteration.

The tabu list is initially empty, constructed in the firstit-erations ( : tabu list size) and updated in latter iterations. Thetabu list ensures that the tabu search avoids returning to the so-lutions just visited and since degrading moves are allowed, thetabu search algorithm has a chance of leaving a local minimum[3].

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736 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 15, NO. 2, MAY 2000

Special attention must be given to the tabu list size. If the tabulist size is too small, the search process may start cycling and ifit is too large, the search may be too restrictive. Therefore, anappropriate list size has to be determined by noting the occur-rence of cycling when the size is too small and the quality of thesolution when the size is too large [3].

The aspiration criterion is usually introduced to speed up thetabu search process. The aspiration criterion temporarily over-rides the tabu status of a move if it is sufficiently good. Thesimplest aspiration criterion is to override the tabu status of amove and allow it into the tabu list if it leads to a better solutionthat the best obtained thus far [4].

Tabu search algorithms are more likely to converge to globaloptima than conventional optimization techniques and can alsotolerate discontinuities, nonlinearities and noisy function eval-uations.

III. PROBLEM FORMULATION AND RESULTS

In this section, the eigenvalue-based objective function usedto robustly select the PSS parameters is formulated, and the op-timization problem solved with a tabu search algorithm. In atypical run of the TS, an initial solution is randomly generated.This initial solution is also assumed to be the current best solu-tion. The current solution is perturbed with moves to generate anew set of trial solutions. Each move generates a trial solution.The number of moves to be made depends on the problem beingconsidered. The move that generates the best solution among theset of trial solutions is selected. This move called the candidatemove is checked to see if it is tabu. If it is not in the tabu list, itbecomes the current solution. If the candidate solution is foundto be tabu, its aspiration criterion is checked. If it passes theaspiration criterion, then it becomes the current best solution,otherwise moves are regenerated to get another set of new solu-tions and the process is repeated. The tabu search is terminatedas soon as the stopping criteria are satisfied [3]. The tabu searchalgorithm is shown in Appendix II.

Consider the problem of determining the parameters of a sta-bilizer that relatively stabilizes a family of plants:

(1)

where is the state vector and is thecontrol vector.

Very often, the closed-loop modes are specified to have somedegree of relative stability. In this case, the closed-loop eigen-values are constrained to lie to the left of a vertical line corre-sponding to a specified damping factor.

A necessary and sufficient condition for the set of plants ineqn. (1) to be simultaneously relatively stabilizable with a singlecontrol law is that the eigenvalues of the closed-loop system liein the left-hand side of a vertical line in the complex-plane[provided a stabilizing parameters set exists]. This conditionmotivates the following approach for determining the param-eters of the PSS.

Select the parameters of the PSS to minimize the followingobjective function:

Re (2)

Fig. 1. Region in the left-hand side of a vertical line.

where is the closed-loop eigenvalue of the plant,subject to the constraints that finite bounds are placed on thestabilizer parameters. The relative stability is determined by thevalue of as shown in Fig. 1. Plainly, if a solution is foundsuch that , then the resulting parameters simultaneouslyrelatively-stabilize the collection of plants. The existence of asolution is verified numerically by minimizing.

IV. SYSTEM MODEL

In this study, two power systems are considered. The firstsystem is a single-machine-infinite bus power system [9]. Thesecond system is the 10-machine 39-bus New England powersystem [20], with each synchronous machine described by anonlinear fourth-order model as given in the Appendix. Thesupplementary stabilizing signal considered is one proportionalto speed. A widely used conventional PSS is consideredthroughout the study [9]. The transfer function of the ith PSS is:

(3)

The structure of the PSS considered in (3) has been used bymany researchers to assess and compare the results of theirwork. The first term in (3) is a washout term with a time lag.The second term is a lead compensation to improve the phaselag through the system.

The numerical values of and used in the study aregiven in Appendix I.

The remaining parameters, namely,, , and are as-sumed to be adjustable parameters. The optimization problem,namely, the selection of these PSS parameters is easily and ac-curately solved using a tabu search. For a given operating point,the power system is linearized around the operating point, theeigenvalues of the closed-loop system are computed, and the ob-jective function is evaluated. It is worth mentioning that only thesystem electromechanical modes are incorporated in the objec-tive function. The bounds on the parameters used in the TS aregiven in Appendix I.

V. SIMULATION RESULTS

To evaluate the effectiveness of the proposed technique, twopower systems are studied.

A. Example 1. Single-Machine-Infinite-Bus System

In this part of the study, a single machine connected to infi-nite bus through a transmission line, and operating at differentloading conditions, is considered. The linearized incremental

ABDEL-MAGID et al.: ROBUST TUNING OF POWER SYSTEM STABILIZERS 737

Fig. 2. System block diagram.

model of this system, voltage regulator and exciter included, isshown in Fig. 2.

The constants to , with the exception of , whichis only a function of the ratio of the impedance, are dependentupon the actual real and reactive power loading as well as theexcitation levels in the machine [7]. The system data are givenin Appendix I.

The simultaneous damping enhancement of the system isdemonstrated by considering five different loading conditions( ). The operating points were selected randomly asfollows:

The eigenvalues of the system at the five operating points con-sidered, without PSS, are:

The damping ratios of the electromechanical modes are respec-tively:

We used the objective function with the TS and toshift the electromechanical mode of each of the five systems tothe left of the vertical line defined by The gains ofthe PSS and in this case were found to be:

and

The eigenvalues of the five systems, with PSS, are given inTable I:

The damping ratios of the electromechanical modes are:

TABLE IEIGENVALUES WITH PSS [EXAMPLE 1]

Fig. 3. Single line diagram for the New England System [20].

indicating a simultaneous improvement in the response of thefive systems.

The extension of the suggested technique to the multimachinecase is demonstrated next. In general, for a system consistingof machines each equipped with a stabilizer of the type de-scribed in this paper, the number of parameters to be optimizedusing the tabu search and objective functionwill be , andthe number of electromechanical modes to be shifted will be

. If operating points are selected for simultaneous stabi-lization, the parameters will be tuned such that themodes are relocated as specified.

B. Example 2. Thirty Nine-Bus Ten-Machine System

In this part of the study, the 10-machine 39-bus power systemshown in Fig. 3 is considered. Details of the system data aregiven in [20]. Generator is an equivalent power source repre-senting parts of the U.S.-Canadian interconnection system. This

738 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 15, NO. 2, MAY 2000

TABLE IIELECTROMECHANICAL MODES; DAMPING RATIOS [EXAMPLE 2, NO PSSS]

TABLE IIIPSS PARAMETERS [EXAMPLE 2, WITH J ]

large system was selected to further demonstrate the versatilityof the suggested technique.

Without power system stabilizers, the system damping is poorand the system exhibits highly oscillatory response [20]. It istherefore necessary to install one or more PSS to improve the dy-namic performance. To identify the optimum locations of PSS’s,the participation factor method [21] and the sensitivity of PSSeffect method [22] were used. The results indicate that machines

and are the optimum locations for installing PSS’sto damp out the local and inter-areas modes of oscillations.

The robust tuning of the PSS’s is demonstrated by consid-ering three operating points labeled as nominal, light and heavyloading conditions. These loading conditions were obtained byvarying the load admittances of the system.

The system nine electromechanical modes and the dampingratio without PSS’s, for the three operating points considered,are given in Table II. Note that the system is unstable for heavyloading.

Using as the objective function with, , the op-timum values of the PSS parameters for the three PSS’s usedare given in Table III.

The objective function achieved is indicatingthat the mechanical modes have been shifted to the left of thevertical line in the complex -plane.

The system electromechanical modes and damping ratios,with the parameters of the PSS’s set as in Table III, are givenin Table IV.

The dynamic responses of the system to a 6-cycle three-phasefault near bus 29 at the end of line 26–29 are shown in Figs.4–6, for the three loading conditions, and with parameters ofthe PSS’s set as in Table III. The robustness of the PSS is quite

TABLE IVELECTROMECHANICAL MODES; DAMPING RATIOS [EXAMPLE 2, WITH J ]

Fig. 4. Response to a three-phase fault with nominal conditions [New Englandsystem:N = 3, objective functionJ ].

evident. An excellent improvement in the damping over a widerange of operating conditions have been achieved with one setof PSS parameters.

VI. CONCLUSION

The use of tabu search to design robust power system sta-bilizers for power systems working at various operating condi-tions is investigated in this paper. The problem of selecting thePSS parameters, which simultaneously improve the damping atvarious operating conditions, is converted to an optimizationproblem with an eigenvalue-based objective function which issolved by a tabu search algorithm. An objective function is pre-sented allowing the robust selection of the stabilizer parame-ters that will optimally place the closed-loop eigenvalues in theleft-hand side of a vertical line in the complex-plane.

The suggested robust design technique was successfullydemonstrated on a single-machine infinite bus system, and amultimachine power system, operating at different loadingconditions.

The performance of the robust PSS’s, tuned using the sug-gested technique, is verified through eigenvalue analysis andsimulation results.

ABDEL-MAGID et al.: ROBUST TUNING OF POWER SYSTEM STABILIZERS 739

Fig. 5. Response to a three-phase fault with light conditions [New Englandsystem:N = 3, objective functionJ ].

Fig. 6. Response to a three-phase fault with heavy conditions [New Englandsystem:N = 3, objective functionJ ].

APPENDIX ISYSTEM DATA

Single-Machine-Infinite-Bus System

The system data are as follows [9]:

Machine (p.u.)

rad/s sec

sec

Transmission Line (p.u.)

Exciter and Stabilizer

sec sec

sec

sec sec.

bounds for the stabilizer adjustable gain and time constants are[0.01, 10] and [0.03, 1.0], respectively.

Multimachine Power System [20]

Machine Model:

(A.1)

(A.2)

(A.3)

(A.4)

(A.5)

sec sec.

bounds for the stabilizer adjustable gain and time constants are[0.01, 20] and [0.06, 0.8], respectively.

APPENDIX IITABU SEARCH ALGORITHM

The following notation is used in the description of the tabusearch algorithm:

: The set of feasible solutions for a given problem.: Current solution,: Best solution reached: Best solution among a sample of trial solutions.

: Evaluation function of solution: Set of neighborhood of (trial solutions): Sample of neighborhood of;

: Sorted sample in ascending order according to theirevaluation functions,

TL: Tabu list.AL: Aspiration level.

Step(0): Set tabu list (TL) as empty and aspiration level (AL)to be zero.

740 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 15, NO. 2, MAY 2000

Step(1): Set iteration counter . Select an initial solu-tion , and set .

Step(2): Generate randomly a set of trial solutions(neighbor to the current solution) and sort them

in an ascending order, to obtain . Let be thebest trial solution in the sorted set (the first inthe sorted set).

Step(3): If , go to Step(4), else set the bestsolution and go to Step(4).

Step(4): Perform the tabu test. If is NOT in the TL, thenaccept it as a current solution, set , and updatethe TL and AL and go to Step(7), else go to Step(5).

Step(5): Perform the AL test. If satisfied, then override thetabu state, Set , update the AL and go to Step(7),else go to Step(6).

Step(6): If the end of the is reached, go to Step(7),otherwise, let be the next solution in the andgo to Step(4).

Step(7): Perform termination test. If the stopping criterion issatisfied then stop, else Set and go to Step(2).

In our implementation, the search is stopped if either of thefollowing two conditions is satisfied:

• The number of iterations performed since the best solu-tion last changed is greater than a prespecified maximumnumber of iterations; or

• Maximum allowable number of iterations is reached.

REFERENCES

[1] F. Glover and H. J. Greenberg, “New approaches for heuristic search: abilateral linkage with artificial intelligence,”European Journal of Op-erational Research, vol. 39, no. 2, pp. 119–130, 1989.

[2] F. Glover, “Artificial intelligence, heuristic frameworks and tabusearch,”Managerial and Decision Economics, vol. 11, pp. 365–375,1990.

[3] F. Glover, “Tabu search: part 1,”ORSA Journal on Computing, vol. 1,no. 3, pp. 190–206, summer 1989.

[4] F. Glover, “A user’s guide to tabu search,”Annals of Oper. Reas., vol.41, pp. 3–28, 1993.

[5] F.R. Scheif, H.D. Hunkins, G.E. Martin,, and E.E. Hattan, “Excitationcontrol to improve power line stability,”IEEE Transactions on PowerApparatus and Systems, vol. PAS-87, pp. 1426–1434, 1968.

[6] F.P. de Mello and C. Concordia, “Concepts of synchronous machine sta-bility as affected by excitation control,”IEEE Transactions on PowerAppparatus and Systems, vol. PAS-88, pp. 316–327, 1969.

[7] M. K. El-Sherbini and D. M. Mehta, “Dynamic Systems stability - in-vestigation of the effect of different loadings and excitation systems,”IEEE Transactions on Power Appparatus and Systems, vol. PAS-92, pp.1538–1546, 1973.

[8] Y. L. Abdel-Magid and G. W. Swift, “Variable structure power stabilizerto supplement static-excitation system,”PROC. IEE, vol. 123, no. 7, pp.697–701, July 1976.

[9] P.M. Anderson and A.A. Fouad,Power System Control and Sta-bility. Iowa: Iowa State University Press, 1977.

[10] Y.N. Yu, Electric Power System Dynamics. New York: AcademicPress, 1983.

[11] H. A. M. Moussa and Y.N. Yu, “Optimal power system stabilizationthrough excitation and/or governor control,”IEEE Transactions onPower Appparatus and Systems, vol. PAS-91, pp. 1166–1174, 1972.

[12] W.C. Chan and Y.Y. Hsu, “An optimal variable structure power systemstabilizer for power stabilization,”IEEE Transactions on Power Apppa-ratus and Systems, vol. PAS-102, pp. 1738–1746, 1983.

[13] Y.Y. Hsu and C. Y. Hsu, “Design of a proportional-integral power systemstabilizer,” IEEE Transactions on Power Systems, vol. PWRS-1, no. 2,pp. 46–83, 1986.

[14] A. Feliachi and X. Zhanget al., “Power system stabilizers design usingOptimal reduced order models, Part I: Model Reduction,”IEEE Trans-actions on Power Systems, vol. PWRS-3, no. 4, pp. 1670–1675, 1988.

[15] E.V. Larsen and D.A. Swann, “Applying Power System Stabilizers: PartsI, II, and III,” IEEE Transactions on Power Apparatus and Systems, vol.PAS-100, pp. 3017–3046, 1981.

[16] P. Kundur, M. Klein, G.J. Rogers, and M.S. Zwyno, “Application ofpower system stabilizers for enhancement of overall system stability,”IEEE Transactions on Power Systems, vol. 4, pp. 382–388, September1992.

[17] P. Pourbeik and M.I. Gibbard, “Simultaneous coordination of powersystem stabilizers and FACTS device stabilizers in a Multi-Machinepower system for enhancing dynamic performance,”IEEE Transactionson Power Systems, vol. 13, pp. 473–479, 1998.

[18] Y. Abdel-Magid, M. Bettayeb, and M.M. Dawoud, “Simultaneous stabi-lization of power systems using genetic algorithms,”IEE Proc.-Gener.Transm. Distrib., vol. 144, no. 1, pp. 39–44, January 1997.

[19] G.N. Taranto and D.M. Falcao, “Robust decentralized control designusing genetic algorithms in power system damping control,”IEE Proc.-Gener. Transm. Distrib., vol. 145, no. 1, pp. 1–6, January 1998.

[20] T. Hiyama and T. Sameshima, “Fuzzy logic control scheme for on-linestabilization of multimachine power system,”Fuzzy Sets and Systems,vol. 39, pp. 181–194, 1991.

[21] Y. Y. Hsu and C. L. Chen, “Identification of optimum location for sta-bilizer applications using participation factors,” inIEE Proceedings, Pt.C, vol. 134, May 1987, pp. 238–244.

[22] E. Z. Zhou, O. P. Malik, and G. S. Hope, “Theory and method for selec-tion of power system stabilizer location,”IEEE Trans. on Energy Con-version, vol. 6, no. 1, pp. 170–176, 1991.

Y. L. Abdel-Magid (M’74, SM’87) received theB. Sc. (Honors) from Cairo University, Egypt, in1969 and the M. Sc. and the Ph. D. degrees fromthe University of Manitoba, Winnipeg, Canada,in 1972 and 1976 respectively, all in ElectricalEngineering. From 1976 to 1979, he was withManitoba Hydro as a Telecontrol engineer. Duringthe 1990-1991 academic year, he was a visitingscholar at Stanford University, Palo Alto, CA, USA.His research interests include power system controland operation, adaptive and robust control of power

systems, and applications of AI techniques in power systems.

M. A. Abido received the B.Sc. (Honors ) andthe M.Sc. degrees in Electrical Engineering fromMenofia University, Egypt, in 1985 and 1989respectively, and the Ph. D. degree from KingFahd University of Petroleum and Minerals, SaudiArabia, in 1997. His research interests are systemidentification, power system control and applicationsof AI techniques in power systems.

A. H. Mantawy received the B.Sc. (Honors ) andthe M.Sc. degrees in Electrical Engineering from AinShams University, Egypt, in 1982 and 1988 respec-tively, and the Ph. D. degree from King Fahd Uni-versity of Petroleum and Minerals, Saudi Arabia, in1997. His research interests includes the applicationsof AI techniques to power system operation and plan-ning.